Yu. N. Deryugin^{1, 2}, M. V. Vetchinnikov¹, D. A. Shishkanov^{1, 2}

¹FSUE RFNC – VNIIEF (Sarov, Russian Federation)

²National Research Mordovia State University (Saransk, Russian Federation)

Abstract. Peridynamics is a non–local numerical method for solving fracture problems based on integral equations. It is assumed that particles in a continuum are endowed with volume and interact with each other at a finite distance, as in molecular dynamics. The influence function in peridynamic models is used to limit the force acting on a particle and to adjust the bond strength depending on the distance between the particles. It satisfies certain continuity conditions and describes the behavior of non-local interaction. The article investigates various types of influence function in peridynamic models on the example of three-dimensional problems of elasticity and fracture. In the course of the work done, the bond-based and state-based fracture models used in the Sandia Laboratory are described, 6 types of influence functions for the bond-based model and 2 types of functions for the state-based model are presented, and the corresponding formulas for calculating the stiffness of the bond are obtained. For testing, we used the problem of propagation of a spherically symmetric elastic wave, which has an analytical solution, and a qualitative problem of destruction of a brittle disk under the action of a spherical impactor. Graphs of radial displacement are given, raster images of simulation results are shown.

Key Words: peridynamics, molecular dynamics, influence function, bond stiffness function, nonlocal interactions, interaction horizon, bond

For citation: Yu. N. Deryugin, M. V. Vetchinnikov, D. A. Shishkanov. Investigation of different influence functions in peridynamics. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:4(2023), 342–360. DOI: https://doi.org/10.15507/2079-6900.25.202304.342-360

Information about the authors:

Yuriy N. Deryugin, Chief Researcher, Russian Federal Nuclear Center (22 Yunosti St., Sarov 607182, Russia), Dr.Sci. (Phys.-Math.), professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0002-3955-775X, dyn1947@yandex.ru

Maxim V. Vetchinnikov, Head of research laboratory, Russian Federal Nuclear Center (22 Yunosti St., Sarov 607182, Russia), ORCID: https://orcid.org/0000-0003-0321-1738 vetchinnikov_max@mail.ru

Dmitry A. Shishkanov, research laboratory mathematician, Russian Federal Nuclear Center (22 Yunosti St., Sarov 607182, Russia), postgraduate, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0002-3063-4798, dima.shishkanov.96@mail.ru